Irreducible Element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.
Irreducible elements should not be confused with prime elements. (A non-zero non-unit element in a commutative ring is called prime if whenever for some and in, then or .) In an integral domain, every prime element is irreducible, but the converse is not true in general. The converse is true for UFDs (or, more generally, GCD domains.)
Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if is a GCD domain, and is an irreducible element of, then the ideal generated by is an irreducible ideal of .
Read more about Irreducible Element: Example
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